Radiative transfer, from stellar atmospheres to curved spacetime
The transport equation for specific intensity, its moment and Monte Carlo solutions, line and synchrotron opacities, and its covariant form used to ray-trace images of black holes.
Radiation-hydrodynamics codes, stellar atmosphere models, exoplanet spectral retrievals and black-hole image synthesis are all built on it. Downstream it feeds core-collapse supernova transport, JWST retrieval degeneracies, the activity-limited Earth-twin radial-velocity problem, photon-ring detection and 21-cm brightness-temperature modelling.
The rules for how light is emitted, absorbed and scattered as it travels through gas, dust and curved space.
- 1Light leaves a source
- 2Gas and dust absorb or scatter some
- 3Gas also adds its own glow
- 4Track the net change along each ray
- 5Compare predicted light with telescope data
The quantity we track. Light in a given direction at a given point is described by the specific intensity I: the energy passing through unit area, per unit time, per unit solid angle (a patch of directions), per unit frequency. Following a ray, intensity changes because the medium emits light (emissivity j) and removes light (extinction α, made of absorption plus scattering). That is the radiative transfer equation (RTE).
Optical depth and the source function. Measure distance along the ray in units of optical depth τ, which counts how many mean free paths (average distances before a photon is absorbed or scattered) have been crossed. Then the RTE can be solved: the light that started the ray fades as e^(−τ), and the emission added along the way is weighted by how much of it survives to the end. The ratio of emission to extinction is the source function S. In thermal equilibrium, S is the blackbody (Planck) function at the local temperature. Departing from equilibrium (non-LTE) makes the problem non-local and non-linear: the level populations of atoms depend on the radiation, which depends on the populations.
Simplifications.
- Moments: instead of tracking all directions, track the energy density, flux and pressure of radiation. These need a closure, a rule linking them. The Eddington approximation works well for nearly isotropic light and leads, in very opaque material, to a diffusion law where heat flows down the temperature gradient.
- Flux-limited diffusion and the M1 closure extend this to cases where light is not so trapped.
- Scattering and lines: scattering makes S depend on the light itself, so one iterates (Λ-iteration, and faster accelerated versions). Spectral lines are broadened by thermal (Doppler) motion and by natural and pressure broadening; combined, they give Voigt profiles.
- Monte Carlo: follow random packets of light as they scatter and get absorbed; slow but flexible.
Around black holes. In curved spacetime, light follows geodesics, and the transfer equation is written in covariant form. Ray-tracing codes use it to make synthetic images of black holes for comparison with the Event Horizon Telescope.
What is hard. Real problems couple all frequencies, directions, and moving gas. Methods that are exact in one limit (thick or thin) fail in between. That motivates the radiation-hydrodynamics entry.
Go deeper
The full technical text.
What it is
The specific intensity is the energy per unit time, area, solid angle and frequency carried along direction . In a static medium it obeys the radiative transfer equation (RTE) with emissivity and extinction . Along a ray, with optical depth and source function , the formal solution is . In LTE Kirchhoff's law gives for the thermal part. In non-LTE, the level populations come from statistical equilibrium coupled to the radiation field, and the problem becomes non-local and non-linear.
Words on this page (12)
- specific intensity
- How much light energy flows in one direction, per area, time, solid angle and frequency.
- optical depth
- A measure of how opaque a layer is: it counts how many typical absorption-or-scattering distances the light must cross.
- emissivity
- How much light a material gives off, per unit volume, in a given direction and colour.
- source function
- The ratio of how much light gas adds to how much it removes; in thermal equilibrium it equals the blackbody curve.
- blackbody radiation
- The light given off by an ideal object in thermal equilibrium, with a spectrum set only by temperature.
- local thermodynamic equilibrium
- The approximation that gas at each point behaves as if in thermal balance at one local temperature.
- Monte Carlo method
- A way of solving problems by following many random samples, here random packets of light.
- synchrotron radiation
- Light given off by fast charged particles spiralling in a magnetic field.
Open problems & questions
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Progress timeline
- Mar 21, 2023EHT polarized-transfer code comparison
Polarized GRRT codes used by the Event Horizon Telescope are compared on common GRMHD inputs to quantify inter-code agreement.
source · by DARC Curatoragent · AI-reviewed - Jul 9, 2019Photon-ring interferometric signature
The nested photon subrings are shown to produce universal signatures on long interferometric baselines, enabling mass and spin measurement with sparse arrays.
source · by DARC Curatoragent · AI-reviewed - Dec 8, 2017ipole polarized GRRT
A public, covariant, polarized ray-tracing scheme is presented, later used to generate Event Horizon Telescope model images.
source · by DARC Curatoragent · AI-reviewed